Rado图与无限完全图的因子分解
Factorizing the Rado graph and infinite complete graphs
摘要：
设$\mathcal{F}=\{F_α: α\in \mathcal{A}\}$和$Λ$是一族无限图.因子分解问题$FP(\mathcal{F}, Λ)$询问$\mathcal{F}$是否可以实现为$Λ$的因子分解，即，是否存在$Λ$的因子分解$\mathcal{G}=\{Γ_α: α\in \mathcal{A}\}$，使得每个$Γ_α$是$F_α$的副本。当$Λ$是Rado图$R$或无限阶$\aleph$的完全图$K_\aleph$时，我们研究了这个问题.当$\mathcal{F}$是可数族时，我们证明了$FP(\mathcal{F}, R)$是可解的当且仅当$\mathcal{F}$中的每个图都没有有限控制集.证明了当图的基数$\mathcal{F}$与图的阶数和控制数一致时，$FP(\mathcal{F}, K_\aleph)$有解.对于可列完全图，当$\mathcal{F}$中图的控制数有限时，我们给出了一些不存在的结果。更确切地说，我们表明，没有分解的$K_{\mathbb{N}}$到$k$ -星（即顶点不相交工会的$k$可数的明星）的副本时，$k=1,2$，而它存在时，$k\geq 4$，留下的问题开放的$k=3$。最后，我们确定了充分条件的图的分解被安排到决议类。
英文摘要：
Let $\mathcal{F}=\{F_α: α\in \mathcal{A}\}$ be a family of infinite graphs, together with $Λ$. The Factorization Problem $FP(\mathcal{F}, Λ)$ asks whether $\mathcal{F}$ can be realized as a factorization of $Λ$, namely, whether there is a factorization $\mathcal{G}=\{Γ_α: α\in \mathcal{A}\}$ of $Λ$ such that each $Γ_α$ is a copy of $F_α$. We study this problem when $Λ$ is either the Rado graph $R$ or the complete graph $K_\aleph$ of infinite order $\aleph$. When $\mathcal{F}$ is a countable family, we show that $FP(\mathcal{F}, R)$ is solvable if and only if each graph in $\mathcal{F}$ has no finite dominating set. We also prove that $FP(\mathcal{F}, K_\aleph)$ admits a solution whenever the cardinality $\mathcal{F}$ coincide with the order and the domination numbers of its graphs. For countable complete graphs, we show some non existence results when the domination numbers of the graphs in $\mathcal{F}$ are finite. More precisely, we show that there is no factorization of $K_{\mathbb{N}}$ into copies of a $k$-star (that is, the vertex disjoint union of $k$ countable stars) when $k=1,2$, whereas it exists when $k\geq 4$, leaving the problem open for $k=3$. Finally, we determine sufficient conditions for the graphs of a decomposition to be arranged into resolution classes.
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